The Quantum-statistical Theory of Transport Phenomena, III

Abstract

In order to introduce the coarse-grained measurement of coordinates in qunatum mechanics, the whole space is divided into cubic cells, and the coordinate of the center of cell is regarded as the coarse-grained coordinate of every point inside the cell. The function which vanishes outside a certain cell and which agrees with plane wave within the same cell is called a cell function. If one imposes a certain type of boundary condition upon such cell functions, the set of these functions is complete and orthonormal, and gives a generalized type of the Wigner distribution functions, from which the coarse-grained distribution functions are derived The change of the density matrices with time is calculated by means of the method analogous to the time-dependent perturbation theory. From this result the collision terms of the Uehling-Uhlenbeck equation are derived on the basis of the equation of motion for the generalized phase-space distribution function, under the assumption that the maximum range of the intermolecular force is sufficiently short compared with the edge length of elementary cell. Owing to the cell functions employed instead of the plane waves, the distribution functions retain dependence on the sp'atial coordinates irrespective of the postulate of random a priori phases, which plays an important role in derivation of the collision terms. In the present article, this postulate is necessary for derivation of. the streamin~ terms, in which the differentiation operators with respect to the coarse-grained coordinates and the coarse.grained distribution functions appear. The domain of validity of the Uehling-Uhlenbeck equation is discussed, and, in conclusion, it seems doubtful that the Uehling.Uhlenbeck equation is valid even for the transport phenomena in the degenerate Bose gas, on account of the singulartiy of the distribution function in momentum space.